Coordinate Geometry - Conic Section Test 7

Total Questions:50 Total Time: 75 Min

Remaining:

 

Questions 1 of 50

Question:The centre of the conic represented by the equation \(2{x^2} - 72xy + 23{y^2} - 4x - 28y - 48 = 0\) is

Answers Choices:

\(\left( {\frac{{11}}{{15}},\;\frac{2}{{25}}} \right)\)

\(\left( {\frac{2}{{25}},\;\frac{{11}}{{25}}} \right)\)

\(\left( {\frac{{11}}{{15}},\; - \frac{2}{{25}}} \right)\)

\(\left( { - \frac{{11}}{{25}},\; - \frac{2}{{25}}} \right)\)

Questions 2 of 50

Question:The centre of \(14{x^2} - 4xy + 11{y^2} - 44x - 58y + 71 = 0\)

Answers Choices:

(2, 3)

(2, – 3)

(– 2, 3)

(– 2, – 3)

Questions 3 of 50

Question:A parabola passing through the point \(( - 4,\; - 2)\) has its vertex at the origin and y-axis as its axis. The latus rectum of the parabola is

Answers Choices:

6

8

10

12

Questions 4 of 50

Question:The focus of the parabola \({x^2} = - 16y\) is

Answers Choices:

(4, 0)

(0, 4)

(–4, 0)

(0, –4)

Questions 5 of 50

Question:The equation of the latus rectum of the parabola \({x^2} + 4x + 2y = 0\) is

Answers Choices:

\(2y + 3 = 0\)

\(3y = 2\)

\(2y = 3\)

\(3y + 2 = 0\)

Questions 6 of 50

Question:Vertex of the parabola \(9{x^2} - 6x + 36y + 9 = 0\) is

Answers Choices:

\((1/3,\; - 2/9)\)

\(( - 1/3,\; - 1/2)\)

\(( - 1/3,\;1/2)\)

\((1/3,\;1/2)\)

Questions 7 of 50

Question:The length of the latus rectum of the parabola \(9{x^2} - 6x + 36y + 19 = 0\)

Answers Choices:

36

9

6

4

Questions 8 of 50

Question:The axis of the parabola \(9{y^2} - 16x - 12y - 57 = 0\) is

Answers Choices:

\(3y = 2\)

\(x + 3y = 3\)\(2x = 3\)

\(y = 3\)

None of these

Questions 9 of 50

Question:The length of the latus rectum of the parabola \({x^2} - 4x - 8y + 12 = 0\) is

Answers Choices:

4

6

8

10

Questions 10 of 50

Question:The focus of the parabola \(y = 2{x^2} + x\) is

Answers Choices:

(0, 0)

\(\left( {\frac{1}{2},\;\frac{1}{4}} \right)\)

\(\left( { - \frac{1}{4},\;0} \right)\)

\(\left( { - \frac{1}{4},\;\frac{1}{8}} \right)\)

Questions 11 of 50

Question:The point of contact of the tangent \(18x - 6y + 1 = 0\) to the parabola \({y^2} = 2x\)is

Answers Choices:

\(\left( {\frac{{ - 1}}{{18}},\;\frac{{ - 1}}{3}} \right)\)

\(\left( {\frac{{ - 1}}{{18}},\;\frac{1}{3}} \right)\)

\(\left( {\frac{1}{{18}},\;\frac{{ - 1}}{3}} \right)\)

\(\left( {\frac{1}{{18}},\;\frac{1}{3}} \right)\)

Questions 12 of 50

Question:The equation of the common tangent of the parabolas \({x^2} = 108y\) and \({y^2} = 32x\), is

Answers Choices:

\(2x + 3y = 36\)

\(2x + 3y + 36 = 0\)

\(3x + 2y = 36\)

\(3x + 2y + 36 = 0\)

Questions 13 of 50

Question:The angle between the tangents drawn at the end points of the latus rectum of parabola \({y^2} = 4ax\), is

Answers Choices:

\(\frac{\pi }{3}\)

\(\frac{{2\pi }}{3}\)

\(\frac{\pi }{4}\)

\(\frac{\pi }{2}\)

Questions 14 of 50

Question:The line \(y = mx + c\) touches the parabola \({x^2} = 4ay\), if

Answers Choices:

\(c = - am\)

\(c = - a/m\)

\(c = - a{m^2}\)

\(c = a/{m^2}\)

Questions 15 of 50

Question:If \(lx + my + n = 0\) is tangent to the parabola \({x^2} = y\), then condition of tangency is

Answers Choices:

\({l^2} = 2mn\)

\(l = 4{m^2}{n^2}\)

\({m^2} = 4\ln \)

\({l^2} = 4mn\)

Questions 16 of 50

Question:The equation of the tangent to the parabola \({y^2} = 9x\) which goes through the point (4, 10), is

Answers Choices:

\(x + 4y + 1 = 0\)

\(9x + 4y + 4 = 0\)

\(x - 4y + 36 = 0\)

\(9x - 4y + 4 = 0\)

3 and 4 are correct

Questions 17 of 50

Question:The point on the parabola \({y^2} = 8x\) at which the normal is inclined at 60o to the x-axis has the co-ordinates

Answers Choices:

\((6,\; - 4\sqrt 3 )\)

\((6,\;4\sqrt 3 )\)

\(( - 6,\; - 4\sqrt 3 )\)

\(( - 6,\;4\sqrt 3 )\)

Questions 18 of 50

Question:The slope of the normal at the point \((a{t^2},\;2at)\) of the parabola \({y^2} = 4ax\), is

Answers Choices:

\(\frac{1}{t}\)

t

t

\( - \frac{1}{t}\)

Questions 19 of 50

Question:Equation of any normal to the parabola \({y^2} = 4a(x - a)\) is

Answers Choices:

\(y = mx - 2am - a{m^3}\)

\(y = m\,(x + a) - 2am - a{m^3}\)

\(y = m\,(x - a) + \frac{a}{m}\)

\(y = m\,(x - a) - 2am - a{m^3}\)

Questions 20 of 50

Question:Tangents drawn at the ends of any focal chord of a parabola \({y^2} = 4ax\) intersect in the line

Answers Choices:

\(y - a = 0\)

\(y + a = 0\)

\(x - a = 0\)

\(x + a = 0\)

Questions 21 of 50

Question:For the above problem, the area of triangle formed by chord of contact and the tangents is given by

Answers Choices:

8

\(8\sqrt 3 \)

\(8\sqrt 2 \)

None of these

Questions 22 of 50

Question:The point on parabola \(2y = {x^2}\), which is nearest to the point (0, 3) is

Answers Choices:

(\( \pm \) 4, 8)

\(( \pm 1,\,1/2)\)

(\( \pm \) 2, 2)

None of these

Questions 23 of 50

Question:The equation of the ellipse whose centre is at origin and which passes through the points (-3, 1) and (2, -2) is

Answers Choices:

\(5{x^2} + 3{y^2} = 32\)

\(3{x^2} + 5{y^2} = 32\)

\(5{x^2} - 3{y^2} = 32\)

\(3{x^2} + 5{y^2} + 32 = 0\)

Questions 24 of 50

Question:If the eccentricity of an ellipse be 5/8 and the distance between its foci be 10, then its latus rectum is

Answers Choices:

39/4

12

15

37/2

Questions 25 of 50

Question:The lengths of major and minor axis of an ellipse are 10 and 8 respectively and its major axis along the y-axis. The equation of the ellipse referred to its centre as origin is

Answers Choices:

\(\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{{16}} = 1\)

\(\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{25}} = 1\)

\(\frac{{{x^2}}}{{100}} + \frac{{{y^2}}}{{64}} = 1\)

\(\frac{{{x^2}}}{{64}} + \frac{{{y^2}}}{{100}} = 1\)

Questions 26 of 50

Question:If the centre, one of the foci and semi-major axis of an ellipse be (0, 0), (0, 3) and 5 then its equation is

Answers Choices:

\(\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{25}} = 1\)

\(\frac{{{x^2}}}{{25}} + \frac{{{y^2}}}{{16}} = 1\)

\(\frac{{{x^2}}}{9} + \frac{{{y^2}}}{{25}} = 1\)

None of these

Questions 27 of 50

Question:The foci of \(16{x^2} + 25{y^2} = 400\) are

Answers Choices:

\(( \pm 3,\;0)\)

\((0,\; \pm 3)\)

\((3,\; - 3)\)

\(( - 3,\;3)\)

Questions 28 of 50

Question:Eccentricity of the ellipse \(9{x^2} + 25{y^2} = 225\) is

Answers Choices:

\(\frac{3}{5}\)

\(\frac{4}{5}\)

\(\frac{9}{{25}}\)

\(\frac{{\sqrt {34} }}{5}\)

Questions 29 of 50

Question:The equation of ellipse whose distance between the foci is equal to 8 and distance between the directrix is 18, is

Answers Choices:

\(5{x^2} - 9{y^2} = 180\)

\(9{x^2} + 5{y^2} = 180\)

\({x^2} + 9{y^2} = 180\)

\(5{x^2} + 9{y^2} = 180\)

Questions 30 of 50

Question:In an ellipse the distance between its foci is 6 and its minor axis is 8. Then its eccentricity is

Answers Choices:

\(\frac{4}{5}\)

\(\frac{1}{{\sqrt {52} }}\)

\(\frac{3}{5}\)

\(25{x^2} + 144{y^2} = 900\)

Questions 31 of 50

Question:The co-ordinates of the foci of the ellipse \(3{x^2} + 4{y^2} - 12x - 8y + 4 = 0\) are

Answers Choices:

(1, 2), (3, 4)

(1, 4), (3, 1)

(1, 1), (3, 1)

(2, 3), (5, 4)

Questions 32 of 50

Question:The eccentricity of the curve represented by the equation \({x^2} + 2{y^2} - 2x + 3y + 2 = 0\) is

Answers Choices:

0

1/2

\(1/\sqrt 2 \)

\(\sqrt 2 \)

Questions 33 of 50

Question:If any tangent to the ellipse \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\)cuts off intercepts of length h and k on the axes, then \(\frac{{{a^2}}}{{{h^2}}} + \frac{{{b^2}}}{{{k^2}}} = \)

Answers Choices:

0

1

1

None of these

Questions 34 of 50

Question:If the line \(y = mx + c\)touches the ellipse \(\frac{{{x^2}}}{{{b^2}}} + \frac{{{y^2}}}{{{a^2}}} = 1\), then \(c = \)

Answers Choices:

\( \pm \sqrt {{b^2}{m^2} + {a^2}} \)

\( \pm \sqrt {{a^2}{m^2} + {b^2}} \)

\( \pm \sqrt {{b^2}{m^2} - {a^2}} \)

\( \pm \sqrt {{a^2}{m^2} - {b^2}} \)

Questions 35 of 50

Question:The value of \(\lambda \), for which the line \(2x - \frac{8}{3}\lambda y = - 3\) is a normal to the conic \({x^2} + \frac{{{y^2}}}{4} = 1\) is

Answers Choices:

\(\frac{{\sqrt 3 }}{2}\)

\(\frac{1}{2}\)

\( - \frac{{\sqrt 3 }}{2}\)

\(\frac{3}{8}\)

Questions 36 of 50

Question:The pole of the straight line \(x + 4y = 4\) with respect to ellipse \({x^2} + 4{y^2} = 4\) is

Answers Choices:

(1, 4)

(1, 1)

(4, 1)

(4, 4)

Questions 37 of 50

Question:The equation of the hyperbola whose conjugate axis is 5 and the distance between the foci is 13, is

Answers Choices:

\(25{x^2} - 144{y^2} = 900\)

\(144{x^2} - 25{y^2} = 900\)

\(144{x^2} + 25{y^2} = 900\)

\(25{x^2} + 144{y^2} = 900\)

Questions 38 of 50

Question:The length of the transverse axis of a hyperbola is 7 and it passes through the point (5, -2). The equation of the hyperbola is

Answers Choices:

\(\frac{4}{{49}}{x^2} - \frac{{196}}{{51}}{y^2} = 1\)

\(\frac{{49}}{4}{x^2} - \frac{{51}}{{196}}{y^2} = 1\)

\(\frac{4}{{49}}{x^2} - \frac{{51}}{{196}}{y^2} = 1\)

None of these

Questions 39 of 50

Question:The locus of the centre of a circle, which touches externally the given two circles, is

Answers Choices:

Circle

Parabola

Hyperbola

Ellipse

Questions 40 of 50

Question:The foci of the hyperbola \(2{x^2} - 3{y^2} = 5\), is

Answers Choices:

\(\left( { \pm \frac{5}{{\sqrt 6 }},\;0} \right)\)

\(\left( { \pm \frac{5}{6},\;0} \right)\)

\(\left( { \pm \frac{{\sqrt 5 }}{6},\;0} \right)\)

None of these

Questions 41 of 50

Question:Centre of hyperbola \(9{x^2} - 16{y^2} + 18x + 32y - 151 = 0\) is

Answers Choices:

(1, –1)

(–1, 1)

(–1, –1)

(1, 1)

Questions 42 of 50

Question:The equation of the hyperbola whose foci are (6, 4) and (-4, 4) and eccentricity 2 is given by

Answers Choices:

\(12{x^2} - 4{y^2} - 24x + 32y - 127 = 0\)

\(12{x^2} + 4{y^2} + 24x - 32y - 127 = 0\)

\(12{x^2} - 4{y^2} - 24x - 32y + 127 = 0\)

\(12{x^2} - 4{y^2} + 24x + 32y + 127 = 0\)

Questions 43 of 50

Question:The equation of the tangent to the hyperbola \(2{x^2} - 3{y^2} = 6\)which is parallel to the line \(y = 3x + 4\), is

Answers Choices:

\(y = 3x + 5\)

\(y = 3x - 5\)

\(y = 3x + 5\) and \(y = 3x - 5\)

None of these

Questions 44 of 50

Question:The locus of the point of intersection of any two perpendicular tangents to the hyperbola is a circle which is called the director circle of the hyperbola, then the equation of this circle is

Answers Choices:

\({x^2} + {y^2} = {a^2} + {b^2}\)

\({x^2} + {y^2} = {a^2} - {b^2}\)

\({x^2} + {y^2} = 2ab\)

None of these

Questions 45 of 50

Question:Let E be the ellipse \(\frac{{{x^2}}}{9} + \frac{{{y^2}}}{4} = 1\) and C be the circle \({x^2} + {y^2} = 9\). Let P and Q be the points (1, 2) and (2, 1) respectively. Then

Answers Choices:

Q lies inside C but outside E

Q lies outside both C and E

P lies inside both C and E

P lies inside C but outside E

Questions 46 of 50

Question:The length of the chord of the parabola \({y^2} = 4ax\) which passes through the vertex and makes an angle \(\theta \) with the axis of the parabola, is

Answers Choices:

\(4a\cos \theta \,{\rm{cose}}{{\rm{c}}^2}\,\theta \)

\(4a{\cos ^2}\theta \,{\rm{cosec}}\,\theta \)

\(a\cos \theta \,{\rm{cose}}{{\rm{c}}^2}\,\theta \)

\(a{\cos ^2}\theta \,{\rm{cosec}}\,\theta \)

Questions 47 of 50

Question:The locus of the point of intersection of lines \((x + y)t = a\) and \(x - y = at\), where t is the parameter, is

Answers Choices:

A circle

An ellipse

A rectangular hyperbola

None of these

Questions 48 of 50

Question:The equation of the hyperbola referred to its axes as axes of coordinate and whose distance between the foci is 16 and eccentricity is \(\sqrt 2 \), is

Answers Choices:

\({x^2} - {y^2} = 16\)

\({x^2} - {y^2} = 32\)

\({x^2} - 2{y^2} = 16\)

\({y^2} - {x^2} = 16\)

Questions 49 of 50

Question:The eccentricity of the hyperbola \(\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{{25}} = 1\) is

Answers Choices:

3/4

3/5

\(\sqrt {41} /4\)

\(\sqrt {41/5} \)

Questions 50 of 50

Question:The equation to the hyperbola having its eccentricity 2 and the distance between its foci is 8

Answers Choices:

\(\frac{{{x^2}}}{{12}} - \frac{{{y^2}}}{4} = 1\)

\(\frac{{{x^2}}}{4} - \frac{{{y^2}}}{{12}} = 1\)

\(\frac{{{x^2}}}{8} - \frac{{{y^2}}}{2} = 1\)

\(\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{9} = 1\)